Fundamental Theorem of Calculus
The fundamental theorem of calculus is a theorem that likes the concept of the derivative of a function with the concept of the integral.
(i) First fundamental theorem of calculus:
Let f be a continuous real valued function defined on a closed interval [a, b]. let F be the function defined, for all x in [a, b] by \(F(x)=\int\limits_{a}^{x}{f(t).dt}\).
Then, F is continuous on [a, b], differentiable on the open interval (a, b) and F’(x) = f(x) for all x in (a, b).
(ii) Second fundamental theorem of calculus/ Newton – Leibnitz Axiom:
Let f and g be real valued function defined on a closed interval [a, b] such that the derivative of g is f. i.e., f and g are fundamental such that for all x in [a, b].
f (x) = g’ (x)
If f is Riemann integrable on [a, b], then \(\int\limits_{a}^{x}{f(x).dx=}g(b)-g(a)\).
Example: \(\int\limits_{0}^{1}{{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right).dx}\) is equal to
Solution: Given,
\(\int\limits_{0}^{1}{{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right).dx}\).
Put tanθ = x → sec²θ dθ = dx (or) θ = tan⁻ˡ(x)
For limit when x = 0 → θ = 0
When x = 1→ θ = π/4
\(I=\int\limits_{0}^{\frac{\pi }{4}}{{{\sin }^{-1}}\left( \frac{2\tan \theta }{1+{{\left( \tan \theta \right)}^{2}}} \right).{{\sec }^{2}}\theta .d\theta }\),
\(I=\int\limits_{0}^{\frac{\pi }{4}}{{{\sin }^{-1}}\left( \sin (2\theta ) \right).{{\sec }^{2}}\theta .d\theta }\) (∴ \(\frac{2\tan \theta }{1+{{\left( \tan \theta \right)}^{2}}}=\sin \left( 2\theta \right)\)),
\(I=\int\limits_{0}^{\frac{\pi }{4}}{2\theta .{{\sec }^{2}}\theta .d\theta }\).
Now applying rule of integrated by parts.
\(I=2\left[ \theta \int{{{\sec }^{2}}\theta -\int{\left\{ \left( \frac{d}{d\theta }\theta \right) \right\}\int{{{\sec }^{2}}\theta .d\theta }}} \right]_{0}^{\frac{\pi }{4}}\),
\(2\left[ \theta \tan \theta -\int{1.\tan \theta .d\theta } \right]_{0}^{\frac{\pi }{4}}\),
\(2\left[ \theta \tan \theta -\log \left| \cos \theta \right| \right]_{0}^{\frac{\pi }{4}}\),
\(2\left[ \frac{\pi }{4}.\tan \left( \frac{\pi }{4} \right)+\log \left| \cos \left( \frac{\pi }{4} \right) \right| \right]-2\left( 0+\log \left| \cos 0 \right| \right)\),
\(\frac{\pi }{2}+2\log \left( \frac{1}{\sqrt{2}} \right)\),
\(\frac{\pi }{2}+2\log {{2}^{(-1/2)}}\),
\(\frac{\pi }{2}+2\times \left( \frac{-1}{2} \right)\log 2\),
\(\frac{\pi }{2}+2\log {{2}^{(-1/2)}}\).